Negate a Statement in Contrapositive Form

Question

Consider the following statement:

If $f_1(x_1^*) = f_n(x_n^*)$ for all $n \geq 1$, then $x_1^* = x_n^* = 1$ for all $n \geq 1$.

I would like to write down the contrapositive statement. Here is my thinking:

Suppose that there exists an $n \geq 1$ such that $x_n^* \neq 1$, then $f_1(x_1^*) \neq f_n(x_n^*)$.

My question is the following: Do I really need to include the quantifier for the statement after "then"? ;i.e., do I have to say $f_1(x_1^*) \neq f_n(x_n^*)$ for some $n$ ? If so, then I think I have "two $n$" flows around which confuses me a lot. Or can I just recycle the $n$ used in the sentence of "suppose..." Is this the correct way to think of?

Any suggestion is appreciated.

Answer 1

If $f_1(x_1^*) = f_n(x_n^*)$ for all $n \geq 1$, then $x_1^* = x_n^* = 1$ for all $n \geq 1$.

It's clear here that the two $n$'s refer to different $n$, as for both statements it's explained that they hold for any $n$. Nevertheless, you repeated the quantifier, as you should.

When you make the contrapositive, you should especially also repeat the quantifier to make sure the reader understands we're talking about different $n$, especially because in this case we're considering specific values of $n$:

Suppose that there exists an $n \geq 1$ such that $x_n^* \neq 1 \color{red}{\vee x^*_1\neq 1}$, then there exists an $n \geq 1$ such that $f_1(x_1^*) \neq f_n(x_n^*)$.

Even better would be to use a different variable to completely remove ambiguity.

Answer 2

You cannot recycle the $n$. Try replacing the $n$ in the first statement with $m$, and it still retains the same meaning. In other words, they are two separate variables in two separate scopes, but with the same name. That's bad practice, precisely because of the confusion you have now.

Answer 3

Your original statement is $p \to q$. You want $\neg q \to \neg p$.

Not knowing anything else about your statements, I'd think $\neg q$ is

"$x_1^* \neq 1$ or $x_n^* \neq 1$ for some $n \geq 1$",

And $\neg p$ would be

"$f_1(x_1^*) \neq f_n(x_n^*)$ for some $n \geq 1$".